You can define 'thanksgiving' broadly to include hymns of praise suited to the season. Of course that probably sounds like what you choose for a recessional hymn. So that provides an approach: move the recessional hymn to this post-communion slot, sing 3 or 4 verses of it, and then play an instrumental recessional.

We chant the Anima Christi.

At our Missa Cantata we sing the Communion antiphon with verses while the people are receiving and follow with an acapella setting of a brief Eucharistic hymn.I'm always looking for new settings of O Salutaris Hostia for this purpose and we have quite a collection already.

I never tire of saying, the Magnificat is the consummate communion canticle.

And, if one is faced with a progressive priest demanding a hymn of thanksgiving, defaulting to versions of the Magnificat could be buttressed by suggestions such as:

1. It's Scripture (if it's not a metrical paraphrase; it's "Scripture-based" if it's a (good) paraphrase). Vatican II valorized opening up for the faithful the riches of Scripture in the liturgy.

2. It's a Scriptural quote of ... a woman. (Not just any old woman, mind you ...). How often do we get to hear from a woman in Scripture in the liturgy? Not enough.

3. It's a canticle that exalts God for his exaltation of the lowly and marginal in the world; possibly the most pointed Gospel canticle that promotes the preferential option for the poor. The text perfectly unites communion with God with communion with neighbor, the two dimensions of the two great commandments.

4. Is there any Scriptural canticle that more aptly describes how we should praise God for receiving his very being within our very being?

I could go on, but you should be able to catch the drift and improvise accordingly.

Soul of My Saviour (Anima Christi) was origi ally sung as a thanksgiving antiphon after mass. I usually have it as a second hymn at communion. It's a nice short filler if the communion music hasnt been quite long enough.

I can not decide if I think this Anima Christi - Adoro Te Devote mashup is wonderful or awful.

Years ago I was in a parish where the choir sang during communion, then everyone stood and sang a Hymn of Thanksgiving. The final blessings were given and the recessional was the organ postlude. Since few people sang either the communion song or the final hymn, combining into one hymn which everyone sang while nothing else was going on seemed to work very well. Is anyone doing this now?

Conflating postures, the lections, processions, gestures and the Caesar of Imperialist jingoes, "You vill zing und you vill like it, und now you must stand zusammen (together) vitch iss za ztronger zymbol zan kneeleenk zusammen, do it, do it NOW! (Imagine Schwarzeneggar ordering you), into conceptual equanimity with Holy Eucharist has probably provoked the late Fr. Eugene Walsh and God into having a good laugh on a heavenly cloud with God remarking "Geno, I see what you did there. Sonny's not too thrilled about it, though."

Kathy, I agree. My choir has sung “Jesus, My Lord, My God, My All” and “The King of Love, My Shepherd Is” for Communion after the English antiphon is sung. They are well-known and well-loved, with beautiful melodies that are mellow and calm. There are other hymns that work here, but these are two of the best. We also sing them SATB, which is our way of saying to the congregation, “You can sing if you like, but for many this is a time of quiet prayer aided by the choir.” We do that or we sing a motet (the same at the Offertory, though the hymn is unison because it is so short)...I don’t love the principle of a Communion hymn because I think it ought to be for private thanksgiving, but that is out of my control and I do appreciate a good hymn, even at that moment at Mass...

In the EF, I would say sing a motet and then improvise softly on the organ or just play the organ.